[MUSIC] Ralph Waldo Emerson once said that an ounce of practice is worth more than a ton of theory. There's a lot of truth in that and practice takes understanding, of course, to another level. I've had a lot of students who get excited by a topic. They feel they understand the theory but then they become frustrated because they aren't able to apply what they've learned to solve problems. This classic application challenge is not unique to finance but it's common whenever we develop procedural knowledge. Of course, practice is the key to problem-solving. You can accelerate your applied skills by avoiding common mistakes that beginners make and overcome bottlenecks that you might find yourself stuck in. I'm going to share some insights into this procedural knowledge that I've learned from years of practice which is guaranteed to enhance your problem-solving skills. All you have to do is consistently apply what I'm going to describe in a three-step procedure whenever you practice a problem and with time complex problems will become easier and easier. Let's just get started. Now you've just read a problem. The first step is to visualize that problem. All of this information that's in your head, needs to go on what we call a timeline. Even if the information is qualitative in nature, try and visualize it. Timelines in finance really help you to do that, they minimize the calculation errors and so the first step is the easiest one. You simply draw a timeline and it might look like this. Here it is, start and end. This is a timeline. You've just done step one. In some cases, this timeline is going to just continue on. It won't be discrete. If it continues on forever, perhaps like this, we can just put this infinity sign in front of it to remind us that this is a perpetuity, it just goes on indefinitely. That was step one. Now the second step, the second step is to represent the information that you're given in a problem on this particular timeline. The information will typically describe time periods, they could be days, they could be months, they could be years, it doesn't matter. You just need to specify the periods and then divide up the timeline for the time horizon. Let's assume in a problem we're going to solve right now that you divide the timeline into three parts. Here's my timeline. It's discrete, it has three periods, 1, 2 and 3. [COUGH] Each of these periods depicts a year. Now, suppose the problem states that you're actually making deposits every six months. Now you know there are two, six month intervals within a year. A three year problem will then be depicted in a timeline that has, in fact, six periods. We should really, for six month deposits, have something that looks like this, 1, 2, 3, 4, 5 and 6, this being times 0. To generalize, suppose we capture the frequency of time with the letter m. When the amounts coincide annually, m is going to equal 1. Of course, when it's semi-annual, as we had in this particular problem, n becomes 2. If it's quarterly it will become 4, if it's monthly, 12, daily, 365, you get the picture. So that would take care of time periods. The bottom line is this, given any number of years, the timeline adjusts these years by the frequency of amounts within a year, which is depicted by m, so that time periods are equal to t times m. If we have a three year problem, m is equal to t2, it becomes a six period problem. Remember, the time period is going to be t times m, that's what we want to depict in terms of periods. In our six period problem where, let's assume we're making deposits and the deposits are $100 every six months, how would we show that on a timeline? That's really simple. We are simply going to take those $100 amounts and show them where they happen. I deposit at the end of six months, after another six months, another six months and so on, until I have all my information on the timeline. [COUGH] Now be careful if the amounts are being shown at the beginning of a period or the end of the period. This is really important where a lot mistakes are made. Of course, here, this is at the end of the period. What if the problem stated that you're making deposits at the beginning of each period? What would that look like on a timeline? That's actually quite simple, we would have the same six period timeline, but the first amount would begin here, beginning of six month period, next, beginning of the next six month period, and so on. We would not have the six deposits, so the amounts simply shift closer to 0. But this is important, so you know exactly when the amounts are corresponding to the time periods. One other point, most problems will give you an interest rate. Let's assume in this particular problem, the interest rate is given to be, let's choose one, 10%. The convention, of course, is that rates are expressed as what is known formally as annual percentage rates. This interest rate is for one year, annually, per annum but our problem here stated semi-annual periods because we said that m was equal to 2. Now we need a semi-annual rate. A semi-annual rate simply takes the annual rate and divides it by m. In our example, it's going to be 10% divided by m2, which gives us the semi-annual rate of 5%. [COUGH] Remember now, the interest rate of 5% is now applicable to this six month or semi-annual period. [COUGH] Now, what happens if interest rates are varying over time? If it's varying over time, this will take us to the third and final step to solve the problem. The first two steps help us to visualize the information, the last step now helps us to identify the correct formula we're going to use to solve that problem. Mistakes are made because people are trying to do all of this in their head. But when you start clarifying it and actually looking at where the numbers play out in terms of the timeline, in terms of beginning or end of the period. Whether it's a series of equal numbers, or whether these numbers are a different amount, we can actually look at it and remind ourselves what we need to do. Now, what are we going to do with this series of numbers? Either the problem is going to ask us to convert them into the future or convert them into the present. It's always going to be that. The numbers get converted into the future, or the numbers get converted into the present. If the numbers are going to be converted into the future, either it's going to be one amount that you're converting or a series of numbers that you are converting. The one amount, we refer to as lump sums, and the series, we can refer to as either an annuity, because they are equal amounts at equal time intervals or these are multiple amounts that vary. So far, so good. [COUGH] The three-step procedure for each problem, make it a habit to draw the timeline. Second, transpose the information in terms of its frequency, the interest rates that appropriate for the time period. And then, third you apply the appropriate formula. Now, you might say, what is the appropriate formula? As you're going to see in the videos about time value and other segments for the course, we will give you the formulas. They're only a handful of them. They are not a lot of formulas but the trick would be, of course, is to choose the right tool, apply it, whether it is a future value problem or whether it is a present value problem. So that's it. It's time for you now to practice, practice, and practice.